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Mirrors > Home > ILE Home > Th. List > ax-mulass | GIF version |
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7649. Proofs should normally use mulass 7719 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.) |
Ref | Expression |
---|---|
ax-mulass | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cc 7586 | . . . 4 class ℂ | |
3 | 1, 2 | wcel 1465 | . . 3 wff 𝐴 ∈ ℂ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 1465 | . . 3 wff 𝐵 ∈ ℂ |
6 | cC | . . . 4 class 𝐶 | |
7 | 6, 2 | wcel 1465 | . . 3 wff 𝐶 ∈ ℂ |
8 | 3, 5, 7 | w3a 947 | . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) |
9 | cmul 7593 | . . . . 5 class · | |
10 | 1, 4, 9 | co 5742 | . . . 4 class (𝐴 · 𝐵) |
11 | 10, 6, 9 | co 5742 | . . 3 class ((𝐴 · 𝐵) · 𝐶) |
12 | 4, 6, 9 | co 5742 | . . . 4 class (𝐵 · 𝐶) |
13 | 1, 12, 9 | co 5742 | . . 3 class (𝐴 · (𝐵 · 𝐶)) |
14 | 11, 13 | wceq 1316 | . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)) |
15 | 8, 14 | wi 4 | 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff set class |
This axiom is referenced by: mulass 7719 |
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